EXISTENCE OF ASYMPTOTIC SPEED FOR LEVEL SET EQUATIONS WITH SOURCE TERM (Theory of Evolution Equation and Mathematical Analysis of Nonlinear Phenomena)
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(2) 88 and we give a main result on the asymptotic speed in Section 4. Finally, in Section 5, we discuss some of partial results of qualitative properties of the asymptotic speed. 2. BIRTH AND SPREAD MODEL. In this section, we derive (C) by using the double‐step method, which is considered as a kind of birth and spread model in the theory of crystal growth (see [6, Section 2.6] for details). Consider two initial‐value problems:. (N). \{ begin{ar ay}{l} v_{t}=f(x) in\mathb {R}^{n}\cros (0,\infty), v(\cdot,0)=u_{0} in\mathb {R}^{n}, \end{ar ay}. and. (P). \{ begin{ar ay}{l} w_{t}=(div(\frac{Dw}{|Dw|})+1)|Dw| in\mathb {R}^{n}\cros (0,\infty), w(\cdot,0)=u_{0} in\mathb {R}^{n}. \end{ar ay}. We call (N) and (P) the nucleation problem and the propagation problem, respectively. Define the operators S_{1}(t), S_{2}(t) : Lip (\mathbb{R}^{n})arrow Lip(\mathbb{R}^{n}) , repectively, by S_{1}(t)[u_{0}]:=u_{0}+f(\cdot)t , where. w. and. S_{2}(t)[u_{0}]:=w(\cdot, t) ,. is the unique viscosity solution of (P). For x\in \mathbb{R}^{n}, \tau>0,. i\in N ,. U^{T}(x, i\tau) :=S_{1}(\tau)(S_{2}(\tau)S_{1}(\tau))^{i}[u_{0}] .. (2.1) set (2.2). We call U^{\tau}(x, i\tau) the 7Votter‐Kato product formula. There is a nice general framework for this formula in the theory of viscosity solutions. established by Barles and Souganidis in [2]. If we apply the framework in [2], then we can prove that. iarrow\infty 1\dot{ \imath} mU^{\tau}(x, i\tau)=u(x, t) where. u. locally uniformly for. x\in \mathbb{R}^{n} ,. (2.3). is the viscosity solution of (C).. In light of this, the behavior of u(x,t)/t as. tarrow\infty. can be consider as the behavior of. tar ow\infty1\dot{\imath}m(\lim_{ar ow0}\frac{U^{T}(x,i\tau)}{i\tau}). .. (2.4). The advantage of considering U^{\tau}(x, i\tau) lies in the fact that its graph is a pyramid of finite number of steps. The double‐step method can then be described in a geometrical way as follows:. (N) At each nucleation step, we drop from above an amount of \tau f(x) crystal down to the pyramid with the assumption that the crystals are not sticky;. (P) At each propagation step, each layer of the pyramid evolves under a forced mean curvature flow (V=\kappa+1) , where V is the outward normal velocity and \kappa is its mean curvature in the direction of the outer normal vector..
(3) 89 Let us emphasize that, in general, the growth of the pyramid is highly nonlinear. The reason comes from the fact that the behavior of each layer is extremely complicated (see [5, Section 5]). One particular layer can receive some amount of crystal in each nucleation step, then changes its shape in each propagation step. Of course the layers change not only in a nonlinear way but also in a nonhomogeneous way in each propagation step.. Furthermore, the changes are not monotone (unlike the case. V=1 ).. These affect the next. nucleation step seriously as the receipt of crystals at each layer will change dramatically from time to time. More or less, this says that the problem has double nonlinear effects. 3. LIPSCHITZ ESTIMATE. Let us first recall basic facts which are standard in the theory of viscosity solutions: \bullet. For any given initial data. u_{0}. , which is bounded uniformly continuous, (C) has a. viscosity solution u\in C(\mathbb{R}^{n}\cross[0, \infty)) which is bounded in \mathbb{R}^{n}\cross[0, T] for each T>0. e. The comparison principle for (C) holds.. See a monograph [3] for proofs for instance. In this section, we establish a Lipschitz estimate with respect to. x. for solutions to (C),. which is uniform on t.. Lemma 3.1. For u_{0}\equiv 0 , let T>0 ,. such that for each. u. be the corresponding solution to (C). There exists R_{0}>0. we have. u(x_{T}, s_{T})= \max_{\mathbb{R}^{n}\cross[0,T]}u. for some. (x_{T}, s_{T})\in\overline{B}(0, R_{0})\cross[0, T].. Proof. If f\equiv 0 , then u\equiv 0 and there is nothing to prove. We hence may assume that f\not\equiv 0 . It is clear then that u\geq 0 and u\not\equiv 0. Fix T>0 and set \sigma=\sup_{\mathbb{R}^{n}\cross[0,T]}u>0 . For \varepsilon, \delta>0 sufficiently small, there exists. (x_{\varepsilon,\delta}, t_{\varepsilon,\delta})\in \mathbb{R}^{n}\cross(0, T]. such that. u(x_{\varepsilon,\delta}, t_{\varepsilon,\delta})= \mathbb{R}^{n}\max_{\cros [, ,\tau]}(u(x,t)-\varepsilon t-\delta(|x|^{2}+1)^{1/2})>0. Set. F(p, X) It is clear that. :=. F. −tr. ( I_{n}- \frac{p\otimes p}{|p|^{2} )X)-|p|. for. (p, X)\in(\mathbb{R}^{n}\backslash \{0\})\cross S^{n}.. is degenerate elliptic and F_{*}(0,0)=F^{*}(0,0)=0 , where F_{*},. F^{*}. denote. half‐relaxed limits (see [3] for definition). By the definition of viscosity subsolution, we have. \varepsilon+F(\delta\frac{x_{\varepsilon,\delta}{(|x_{\varepsilon,\delta} |^{2}+1)^{1/2},\delta\frac{(|x_{\varepsilon,\delta}|^{2}+1)I_{n}- x_{\varepsilon,\delta}\otimesx_{\varepsilon,\delta}{(|x_{\varepsilon,\delta}|^ {2}+1)^{3/2})\leqf(x_{\varepsilon,\delta}). ,. where I_{n} is the identity matrix of size n . Let \deltaarrow 0 first to deduce that (x_{\varepsilon,\delta}, t_{\varepsilon,\delta})ar ow(x_{\varepsilon}, t_{\varepsilon}) by passing to a subsequence if necessary and x_{\varepsilon}\in\overline{B}(0, R_{0}) as f=0 on \mathbb{R}^{n}\backslash B(0, R_{0}) . \square We then let \varepsilonarrow 0 to get the desired result..
(4) 90 Lemma 3.2. Let. u. be the solution to (C) with the initial data u_{0}\equiv 0 , Then. \Vert u_{t}\Vert_{L^{\infty}(\mathbb{R}^{n}\cross[0,\infty) }\leq M_{f}, where. M_{f}= \max_{\mathbb{R}^{n}}f.. Proof. It is clear that \varphi(x, t)=M_{f}t for (x, t)\in \mathbb{R}^{n}\cross[0, \infty) is a supersolution to (C) because of the fact that F_{*}(0,0)=F^{*}(0,0)=0 . We use the comparison principle to get. 0\leq u(x, t)\leq M_{f}t. for all (x, t)\in \mathbb{R}^{n}\cross[0, \infty ).. (3.1). Thus, \Vert u_{t}(\cdot, 0)\Vert_{L}\infty(\mathbb{R}^{n})\leq M_{f}. For any given s>0 , both (x, t)\mapsto u(x, t+s) and (x, t)\mapsto u(x, t) are viscosity solutions. to (C) with initial data u(\cdot, s) and u(\cdot, 0) , respectively. By the comparison principle and (3.1),. \Vert u(\cdot, t+s)-u(\cdot, t)\Vert_{L^{\infty}(\mathbb{R}^{n})}\leq\Vert u(\cdot, s)-u(\cdot, 0)\Vert_{L^{\infty}(\mathbb{R}^{n})}\leq M_{f}s. Divide both sides of the above by. Proposition 3.3. Let. u. s. and let. sarrow 0. to get the conclusion.. \square. be the solution to (C) with given initial data u_{0}\equiv 0 . Then, there. exists C>0 such that. \Vert Du\Vert_{L}\infty(\mathbb{R}^{n}\cross[0,\infty) \leq C. Proof. For \varepsilon\in(0,1) , we consider the following approximated equation. \{ begin{ar y}{l u_{t}^\varepsilon}-(div(\frac{Du^{\varepsilon}{\sqrt{|Du^{\varepsilon}1^{2}+ \varepsilon^{2} )+1)\sqrt{|Du^{\varepsilon}|^{2}+\varepsilon^{2}-f=0 in \mathb {R}^{n}\cros (0,\infty), u^{\varepsilon}(x,0)=0 on\mathb {R}^{n}. \end{ar y}. (3.2). This has a unique solution u^{\varepsilon}\in C^{2}(\mathbb{R}^{n}\cross[0, \infty)) . Setting b^{\varepsilon}(p) :=I_{n}-p\otimes p/(|p|^{2}+\varepsilon^{2}) ,. we rewrite (3.2) as. u_{t}^{\varepsilon}-b_{ij}^{\varepsilon}(Du^{\varepsilon})u_{x_{i}x_{j} ^{\varepsilon}-\sqrt{|Du^{\varepsilon}|^{2}+\varepsilon^{2} -f=0. in \mathbb{R}^{n}\cross(0, \infty) .. (3.3). Here we use Einstein’s convention.. We use the Bernstein method to get the gradient bound for u^{\varepsilon} , hence u . Let w^{\varepsilon} := |Du^{\varepsilon}|^{2}/2 . Differentiate the above equation with respect to x_{k} and multiply by u_{x_{k} ^{\varepsilon} to yield. Fix. w_{t}^{\varepsilon}-b_{ij}^{\varepsilon}(w_{x_{i}x_{j} ^{\varepsilon}-u_{x } ^{\varepsilon}ju_{x_{i}x ^{\varepsilon})-Df\cdotDu^{\epsilon}-u_{x_{i}x_{j} ^{ \varepsilon}D_{p}b_{ij}^{\varepsilon}\cdotDw^{\varepsilon}+ \frac{Du^{\varepsilon}\cdotDw^{\varepsilon} {\sqrt{|Du^{\varepsilon}|^{2}+ \varepsilon^{2} =0. T>0 .. we have. Take (x_{0}, t_{0})\in \mathbb{R}^{n}\cross(0, T] so that w^{\varepsilon}(x_{0}, t_{0})= \max_{\mathbb{R}^{n}\cross[0,T]}w^{\varepsilon} . At this point,. b_{ij}^{\varepsilon}u_{x_{j}x_{k} ^{\varepsilon}u_{x_{i}x_{k} ^{\varepsilon}-Df \cdot Du^{\varepsilon}\leq 0 .. (3.4). By using a modified Cauchy‐Schwarz inequality. ( tr AB)^{2}\leq tr(ABB)tr. A. for all A, B\in S^{n}, A\geq 0 ,. (3.5). we obtain. Df\cdotDu^{\varepsilon}\geqtr(b^{\varepsilon}(Du^{\varepsilon})D^{2} u^{\varepsilon}D^{2}u^{\varepsilon})\geq\frac{(tr b^{\varepsilon} (Du^{\varepsilon})D^{2}u^{\varepsilon}) ^{2}{t_{\Gam a}(b^{\varepsilon} (Du^{\varepsilon}) \geq\frac{(tr b^{\varepsilon}(Du^{\varepsilon})D^{2} u^{\varepsilon}) ^{2}{n}. . (3.6).
(5) 91 91 By repeating the proof of Lemma 3.2, we have that. \varepsilon\in(0,1) . We use this and (3.3) to yield. \Vert u_{t}^{\varepsilon}\Vert_{L(\mathbb{R}^{n}\cross[0,\infty) }\infty\leq M_ {f}+1 for all. ( tr (b^{\varepsilon}(Du^{\varepsilon})D^{2}u^{\varepsilon}) ^{2}=(u_{t} ^{\varepsilon}-\sqrt{|Du^{\varepsilon}|^{2}+\varepsilon^{2} -f)^{2}\geq\frac{1} {2}|Du^{\varepsilon}|^{2}-C,. where. (3.7). C=(2M_{f}+1)^{2}.. Combining (3.6) and (3.7) together, we obtain. \frac{1}{2}|Du^{\varepsilon}|^{2}-C\leq nDf\cdot Du^{\varepsilon}\leq C|Du^{\varepsilon}|, which implies that \Vert Du^{\varepsilon}\Vert_{L^{\infty}(\mathbb{R}^{n}\cross[0,\infty) }\leq C for some \Vert Df\Vert_{L}\infty , and n . Let \varepsilonarrow 0 to yield the desired result.. C>0. depending only on \Vert f\Vert_{L}\infty, \square. Remark 1. It is worth emphasizing that the Lipschitz bound which is independent of t in Proposition 3.3 is essential to obtain the existence of the asymptotic speed. Indeed, the time local Lipschitz bound, which can be easily obtained by only using the comparison principle, is not enough. 4. EXISTENCE OF ASYMPTOTIC SPEED. Below is one of main results in [4]. Theorem 4.1. Let. u. be the solution to (C). There exists c\in[0, M_{f}] such that. \lim_{tar ow}\frac{u(x,t)}{t}=c Furthermore,. c. locally uniformly for. is independent of the choice of. x\in \mathbb{R}^{n} .. (4.1). u_{0}.. Proof. Since the comparison principle holds, in order to prove (4.1), we can assume that u_{0}\equiv 0 . Recall that (3.1) gives us. 0=u_{0}(x)\leq u(x, t)\leq M_{f}t. For t\geq 0 , set m(t)= \sup_{x\in \mathbb{R}^{n}}u(x, t) . We now show that. m(t+s)\leq m(t)+m(\mathcal{S}) Fix s\geq 0 . We note that. for all. s,. (x, t)\mapsto v(x, t)=u(x, t+s)-m(s). t\geq 0 . and. (4.2). (x, t)\mapsto u(x, t). are both. solutions to (C), and. v(x, 0)=u(x, s)-m(\mathcal{S})\leq 0=u(x, 0). .. Thus, v(x, t)\leq u(x, t) in light of the comparison principle. In particular, we get that (4.2) holds, which means that m is subadditive on [0, \infty ). By Fekete’s lemma, there exists c\in[0, \infty ) such that. t ar ow\infty 1\dot{ \imath} m\frac{m(t)}{t}=c=s>0\dot{ \imath} nf\frac{m(\mathcal{S}) {s} .. It is clear that c\leq M because of (3.1). If therefore only need to consider the case that. c=0 ,. c>0 .. then (4.1) holds immediately. We \varepsilon>0 . There exists T=T(\varepsilon)>0. Fix. such that. c \leq\frac{m(t)}{t}\leq c+\varepsilon. (4.3). for all t>T..
(6) 92 For. t> \max\{\frac{MT}{c}, M\} ,. we use Lemma 3.1 to have that. ct \leq \mathbb{R}^{n}\cdot\max_{[0,t]}u=u(x_{t}, s_{t})\leq Ms_{t}. for some (x_{t}, s_{t})\in\overline{B}(0, R_{0})\cross[0,t] ,. (4.4). which implies that s_{t} \geq\frac{ct}{M}\geq T . Thus, we are able to improve (4.4) as. ct \leq\max_{\mathbb{R}^{n}\cross[0,t]}u=u(x_{t}, s_{t})\leq(c+\varepsilon) s_{t} , which yields s_{t} \geq\frac{c}{c+\varepsilon}t . So for any x\in B(0, R) for. R>0. (4.5). given, we use Lemma 3.2 and. Proposition 3.3 to estimate that. |u(x, t)-u(x_{t}, s_{t})| \leq C(|x-x_{t}|+|t-s_{t})|\leq C(R+R_{0})+ \frac{C\varepsilon t}{c+\varepsilon}. Hence,. The proof is complete.. c- \frac{C\varepsilon}{c+\varepsilon}-\frac{C(R+R_{0})}{t}\leq\frac{u(x,t)}{t} \leq c+\varepsilon. \square. Remark 2. We note that the use of the Fekete lemma is quite natural in the literature once some subadditive quantities are identified. A related result in the periodic setting. appeared in a lecture note by Barles [1, Theorem 10.2]. In general, the lack of periodicity prevents us from using the natural compactness property of \Gamma^{n} . In a sense, (1.1) is a compactness assumption, which is a simple and effective replacement for the periodicity and also quite natural from viewpoint of physicists. 5. QUALITATIVE PROPERTIES ON ASYMPTOTIC SPEED. In this section, we investigate qualitative properties on asymptotic speed. Let. solution to (C) and we denote the asymptotic speed by on. c_{f}. u. be the. to emphasize the dependence. f.. 5.1. Radial symmetric case. In this subsection, we assume a radially symmetric con‐ dition for f , i.e., f(x)=\tilde{f}(|x|) for x\in \mathbb{R}^{n} , where \tilde{f}:[0, \infty ) arrow \mathbb{R} is given. The following theorem gives a complete characterization of c_{f}.. Theorem 5.1. Assume that. Lip ([0, \infty), [0, \infty)) . Let. u. f(x)=\tilde{f}(|x|). for. x\in \mathbb{R}^{n} ,. where. be the solution to (C) . Then,. \tilde{f}\in C_{c}([0, \infty), [0, \infty))\cap. c_{f}= \max_{r\in[n-1,\infty)}\tilde{f}(r)=\max_{|x|\geq n-1}f(x) .. (5.1). In order to prove this theorem, we consider a radially symmetric solution u(x, t)= \phi(|x|, t) with u(x, 0)=0 for all (x, t)\in \mathbb{R}^{n}\cross[0, \infty). Then \phi=\phi(r, t) : [0, \infty ) \cross[0, \infty ) arrow \mathbb{R} satisfies the 1‐dimensional Hamilton‐Jacobi equation:. \{ begin{ar ay}{l} \phi_{t}-\frac{n-1}{r}\phi_{r}-|\phi_{r}|=\tilde{f}(r) in(0,\infty)\cros (0, \infty), \phi(\cdot,0)=0 on[0,\infty). \end{ar ay}. (5.2).
(7) 93 Lemma 5.2. Set u(x, t) :=\tilde{\phi}(|x|, t) for all (x, t)\in \mathbb{R}^{n}\cross[0, \infty ), where \tilde{\phi}:[0, \infty ) \cross[0, \infty ) \mathb {R}. arrow. is the function defined by. \tilde{\phi}(r,t)=\sup\{\int_{0}^{t}\tilde{f}(\gamma(s) ds : \gamma([0,t]) \subset(0, \infty), \gamma(t)=r, |\gamma'(s)+\frac{n-1}{\gamma(s)}|\leq 1 a.e.\} .. (5.3). Then,. u. is the viscosity solution to (C) .. Notice here that since we consider the viscosity solution (which may not be smooth at x=0) of (C), we do not know the boundary condition of \phi at r=0 a priori. Therefore, it is not clear a priori that we have the representation for. u. in Lemma 5.3. We refer to. [4] for a proof. An important point in this representation is that we have the constraint. | \gamma'(s)+\frac{n-1}{\gamma(s)}|\leq 1. Therefore, for any R\in(0, n-1) , if B(0, R) , then. \gamma. for a.e. \mathcal{S}\in(0, t) .. (5.4). is in the admissible class of (5.3) such that \gamma(s)\in. \gamma'(s)\leq 1-\frac{n-1}{\gamma(s)}\leq 1-\frac{n-1}{R}=-\frac{n-1-R}{R}=:- d<0. This says that the admissible trajectory cannot stay in B(0, n-1) for a long time. From this observation, intuitively, we can see that the asymptotic speed is characterized by. (5.1). We point out here that such a constraint which creates an interesting phenomenon from the view point of the asymptotic speed comes from the noncoercivity of Hamiltonian. in (5.2). We omit the detail of the proof of Theorem 5.1 and refer to [4]. Example 1. Set. f_{r}(x_{1)}x_{2}) := \max\{-(\sqrt{|x_{1}|^{2}+|x_{2}|^{2}})+r, 0\} for r\geq 0 . Set c(r) :=c_{f_{r}} . Due to Theorem 5.1, we obtain. c(r)= \max\{0, r-1\}. for all r\geq 0.. 5.2. Non‐radial symmetric case. It non‐radial symmetric settings, it seems to be much complicated and hard to obtain detailed qualitative properties of the asymptotic speed. We here give some of partial results. By using the comparison principle and Theorem 5.1, we get several results for a general compact set E.. Proposition 5.3. If E\subset B(y, n-1) for some y\in \mathbb{R}^{n} , then. \lim_{tar ow}\frac{u(x,t)}{t}=0 If. \overline{B}(y, n-1)\subset int. E. uniformly for x\in \mathbb{R}^{n}.. for some y\in \mathbb{R}^{n} , then. \lim_{tar ow\infty}\frac{u(x,t)}{t}=c. locally uniformly for. x\in \mathbb{R}^{n}..
(8) 94 Proposition 5.4. Assume there exist \mathcal{S}\in(0, M_{f}) and R<n-1 such that. \{x\in \mathbb{R}^{n} : M_{f}-s\leq f(x)\leq M_{f}\}\subset B(0, R) Let. u. .. be the solution to (C). Then c_{f}\leq M-s.. Proposition 5.5. Assume that there exists R\geq n-1 such that. B(0, R)\subset\{x\in \mathbb{R}^{n} : f(x)=M_{f}\} Let. u. be the solution to (C) . Then c_{f}=M_{f}.. See [5, 4] for proofs of Propositions 5.3−5.5. Example 2. Let R_{0}\in(1/2,1) . Set. f_{a}(x) := \max\{-|(x_{1}, x_{2})-(a, 0)|+R_{0}, -|(x_{1}, x_{2})-(-a, 0)|+ R_{0},0\} for a\geq 0 . Set c(a). :=c_{f_{a}} .. By Proposition 5.3, we have. c(a)=0. if 0<a<1-R_{0},. c(a)=0. if a>R_{0}.. We can easily prove that a\mapsto c(a) is continuous by using the stability of viscosity solutions, and know that c(a)>0 for all a\in(1-R_{0},1) due to [5, Theorem 5.6]. However, we still do not precisely know the shape of c(a) . For instance, it is not clear yet which a gives the maximum of c(a) , which seems to be important from view point of the theory of crystal growth. Acknowledgments. The author would like to thank Professor Ken Shirakawa for the kind invitation to the conference in RIMS, “Theory of Evolution Equation and Mathe‐ matical Analysis of Nonlinear Phenomena 2017” REFERENCES. ı. G. Barles, An introduction to the theory of viscosity solutions for first‐order Hamilton‐Jacobi equations and applications. Hamilton‐Jacobi equations: approximations, numerical analysis and applications, 49‐109, Lecture Notes in Math., 2074, Fond. CIME/ CIME Found. Subser., Springer, Heidelberg, 2013.. 2. G. Barles, P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271‐283. 3. Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99. Birkhäuser, Basel‐Boston‐Berlin (2006), xii+264pp. 4. Y. Giga, H. Mitake, T. Ohtsuka, H. V. Ran, Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations, in preparation. 5. Y. Giga, H. Mitake, H. V. han, On asymptotic speed of solutions to level‐set mean curvature flow equations with driving and source terms, SIAM J. Appl. Math. 48 (2016), no. 5, 3515‐3546. 6. M. Ohara and R. C. Reid, Modeling crystal growth rates from solution. Prentice‐Hall, Englewood Cliffs, N. J. (1973).. (H. Mitake) INSTITUTE OF ENGINEERING, DIVISION OF ELECTRICAL, SYSTEMS AND MATHEMATI‐ ENGINEERING, HIROSHIMA UNIVERSITY 1‐4‐1 KAGAMIYAMA, HIGASHI‐HIROSHIMA‐SHI 739‐8527,. CAL. JAPAN. E‐mail address: hiroyoshi‐mitake@hiroshima‐u.ac.jp.
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